What's your resolution? I assume your thread title says 6970 because you're betting you can flash that 6950 you linked to a 6970? That's not guarunteed, but from what I hear, the success rate seems to be decent.
why did you pick so expensive 6950 and 560? there are much better and cheaper cards... they are way over your budget too, probably 6870 or if you can add 10$ 6950 1GB are in this price range clear winners, also i've heard that you can flash the 1GB version...
At this time there are no distinct performance advantages between the brands.
I keep reading in forums that the HD 6950 pwns the GTX 560 Ti, but when I've looked at pretty much any benchmark the performance between the two is typically within a margin of error that would not be distinguishable by the human eye (1 to 3 frames).
Basically if your price range is under $200, you go and look at the HD 6950, GTX 560 Ti, HD 6970, GTX 470, HD 5870, HD 6870 and which ever one is on sale for $125 to $150, you buy that one.